tsunamiballs: agranularapproachtotsunamirunup …ward/papers/tsunami_balls.pdf2 tsunami balls we...

COMMUNICATIONS IN COMPUTATIONAL PHYSICSVol. 3, No. 1, pp. 222-249

Commun. Comput. Phys.January 2008

Tsunami Balls: A Granular Approach to Tsunami Runup

and Inundation

Steven N. Ward∗and Simon Day

Institute of Geophysics and Planetary Physics, University of California, Santa Cruz,CA 95064, USA.

Received 2 January 2007; Accepted (in revised version) 6 June, 2007

Available online 14 September 2007

Abstract. This article develops a new, granular approach to tsunami runup and in-undation. The small grains employed here are not fluid, but bits, or balls, of tsunamienergy. By careful formulation of the ball accelerations, both wave-like and flood-likebehaviors are accommodated so tsunami waves can be run seamlessly from deep wa-ter, through wave breaking, to the final surge onto shore and back again. In deepwater, tsunami balls track according long wave ray theory. On land, tsunami balls be-have like a water landslide. In shallow water, the balls embody both deep water andon land elements. In modeling several 2-D and 3-D cases, we find that wave breakinggenerally causes relative runup to increase with beach slope and wave period and de-crease with input wave amplitude. Because of their highly non-linear nature, runupand inundation are best considered to be random processes rather than deterministicones. Models and observations hint that for uniform input waves, normalized runupstatistics everywhere follow a single skewed distribution with a spread between 1/2and 2 times its mean.

AMS subject classifications: 76B15

Key words: Water wave, tsunami, run up.

1 Introduction

The thorniest issue in tsunami calculation is not deep water propagation, but rather the”first mile” where the waves generate and the ”last mile” where the waves run into shal-low water and then onto land. Understanding ”first mile” events of tsunami birth bylandslides, asteroid impacts and earthquakes is critical to fixing the intrinsic level of haz-ard. Understanding ”last mile” events of tsunami runup and inundation is critical tomapping site-specific hazard, to the design of wave resistant structures and to the in-terpretation of paleotsunami deposits. This article considers those mysterious last mileevents–tsunami runup and inundation.

∗Corresponding author. Email addresses: [email protected] (S. N. Ward), simonday [email protected]

(S. Day)

http://www.global-sci.com/ 222 c©2008 Global-Science Press

S. N. Ward and S. Day / Commun. Comput. Phys., 3 (2008), pp. 222-249 223

Modeling of water waves initially incident from deep water through shoaling in shal-lows, breaking, and finally runup onshore, must confront the changing nature of thetsunami and the movement of material and energy within. In deep water, tsunami mo-tion is oscillatory and wave-like. The net displacement of the water mass is negligible andthe velocity of energy transmission far exceeds the velocity of the individual water par-ticles. Wave-like behavior makes for efficient propagation of tsunami over transoceanicdistances. As tsunami approach shore, energy concentrates, wave heights increase tobreaking, and water surges onshore. In these final flood-like stages, the net displacementof the water mass becomes large and the velocities of energy transmission and particlemotion equalizes.

Most existing ”last mile” approaches to wave runup and inundation (Synolakis andBernard, 2006) employ non-linear shallow water equations and impose bore-like solu-tions to simulate breaking and post-breaking propagation and energy loss (Peregrine,1966; Hibberd and Peregrine, 1979; Li and Raichlen, 2002). Others employ Boussinesqwave models augmented with extra empirical terms in the equations of motion to at-tempt to account for turbulent energy losses during breaking (Kennedy et al., 2000; Lynettet al., 2002). Developing numerical fluid dynamic models that span the entire ’last mile’however, has proved problematic. Special difficulties lie in quantifying the physics of tur-bulence, especially in proximity to the water-substrate and water-atmosphere interfaces.Those computational fluid dynamic models that do attempt to address runup and in-undation in realistic three dimensional geometries are demanding computationally andsusceptible to numerical instabilities (Herrmann, 2006).

This article develops a new approach to tsunami runup and inundation. The conceptsprings from recent work by Ward and Day, (2006) who simulated the 1980, Mt. St. HelensWashington landslide. In that simulation, granular debris raced down the mountain andthen ran up and deflected from facing slopes much like a bobsled crew vying for a goldmetal. The notion occurred to us that, the on-land phases of tsunami runup and inun-dation with all the vulgarities of deflecting and interacting flows might be modeled as a”water landslide”. In debris avalanches, matter and energy do travel at the same velocity,but basal friction eventually slows and deposits solid landslide material. By eliminatingbasal friction, a water landslide would sooner or later drain back to the sea as does atsunami. Although water landslides might well-emulate the on-land phase of tsunamiinundation, we realized that for the approach to be useful, the deep water and runuptsunami phases have to be addressed too. The challenge posed was, ”Can we formulateappropriate accelerations to carry granular water bits or ’tsunami balls’ seamlessly fromdeep water, to breaking, onto land, and back again?”

2 Tsunami balls

We intend to simulate near shore tsunami wave propagation, runup, and inundation bytracking accelerating tsunami balls. Because the physical behavior of water motion takes

224 S. N. Ward and S. Day / Commun. Comput. Phys., 3 (2008), pp. 222-249

Figure 1: Geometry for accelerating tsunami balls. Quantities are measured as a vector function of time p=p(t)or as a scalar function of distance along the path p = p(t). (Left) In water, tsunami balls will speed or slowdepending on the sign of v(t)•∇H(p(t)) and always accelerate toward shallow water, increasing or decreasingthe azimuth Θ(t) of the path depending on the sign of (v(t)× z)•∇H(p(t)). (Right) On land, we introducethe unit upward normal to the topographic surface n and the down slope direction s (insert) Tsunami balls onland always accelerate down slope.

different forms at different stages in this sequence, the accelerations take three forms–deep water, near shore/breaking and onshore–depending upon the ball’s current loca-tion. Fig. 1 sets the stage for tsunami ball theory with x and y directions being east andnorth in the horizontal plane and z being up so g = −gz. Let T(x,y) be topographicheight measured positive upward from still water. Over water T(x,y)=−H(x,y), whereH(x,y)>0 is the ocean depth.

2.1 Deep water accelerations

In water of depth greater Acrit (to be specified later) we would like tsunami balls to mimiclong water waves. Accordingly, we assume that deep-water tsunami balls at position p(t)that have velocity v(t)=v(t)v(t) in the x-y plane accelerate according to

adw(t)=(g/2)v(t)v(t)•∇H(p(t))−(v(t)× z)v(t)R(t),

R(t)=(g/2v(t))(v(t)× z)•∇H(p(t)).(2.1)

The first term in (2.1) acts in the direction of motion v and speeds or slows the tsunamiball. The second term in (2.1) acts perpendicular to the direction of motion and deflectsthe ball’s path in the horizontal plane without changing its speed. R(t) is the rotationrate of the ball’s direction.

Eq. (2.1) is not meant to represent an equation of motion in the usual sense of conser-vation of momentum; rather, it has been devised to impart on the tsunami balls certain


shallow water wave properties. For instance, if scalar p= p(t) measures distance along atsunami ball path, then v(t)•∇H(p(t))=∂H(p)/∂p and the first term in (2.1) is

a(t)=∂v(t)/∂t=(∂p/∂t)(∂v(p)/∂p)

=v(p)(∂v(p)/∂p)=(g/2)∂H(p)/∂p, (2.2)

where H(p) is the water depth. It is easy to show that v(p)=√

gH(p) satisfies (2.2)

(∂v(p)/∂p)=(g/2v(p))∂H(p)/∂p

so once a tsunami ball is started in any direction with vo =√

gHo, the first term in (2.1)

assures that it will accelerate to keep the long wave tsunami speed of v(p) =√

gH(p)everywhere along its path. Additionally, because

v(p)=√

gH(p), ∇v(p)=(g/2v(p))∇H(p),

the second term in (2.1) is

a(t)=v(p)(∂v(p)/∂p)=−( v(t)× z)(g/2)(v(t)× z)•∇H(p(t))

or(∂v(p)/∂p)=−(v(p)× z)(v(p)× z)•∇v(p). (2.3)

Because this acceleration only changes the direction of v(p) not its magnitude, (2.3) is

∂Θ(p)/∂p=(v(p)× z)•∇v(p)

v(p), (2.4)

where Θ(p) is the angle of the ball’s direction measured from x to y. Eq. (2.4) is justthe standard formula for tracing tsunami rays in a plane (Chou, 1970). Tsunami ballsaccelerating according to (2.1) maintain long wave tsunami speeds and track tsunamipaths as given by ray theory.

Energy balls

In deep water, tsunami are wave-like in that tsunami energy travels at v(p) =√

gH(p)and not the water mass itself. Accordingly, we consider the tsunami balls to possess afixed energy Eo. At any location, the amplitude of the tsunami wave above still waterA(p) relates to the number density σ(p) [m−2 units] of tsunami balls in the x-y plane by

A(p)=

[

σ(p)E0

ρg

]1/2

, (2.5)

where ρ is the density of water. We take the square root in (2.5) to be positive, althoughthe concept of wave troughs being represented by the negative sign choice has been con-sidered.


Imagine now, shooting off one tsunami ball per second governed by (2.1) with v(po)=√

gH(po) in some fixed direction. Because all the balls follow the same path at the samerate and can’t stop, for any other point along the path the balls must also pass one persecond. Energy flux is conserved. Ball speeds change along the path however, so energydensity ρ(p)Eo and wave amplitude (2.5) will change. At the start of the path σ(po)Eo =Eo/[v(po)× (1s) (1m)] and somewhere else along the path σ(p1)Eo = Eo/[v(p1)× (1s)(1m)]. From (2.5), tsunami amplitude at p1 relates to amplitude at po by

A(p1)= A(po)

[

v(po)

v(p1)

]1/2

= A(po)

[

Ho

H1

]1/4

. (2.6)

(2.6) is the shoaling relation (Green’s Law) of linear, long wave tsunami wave theory.

Recapitulation of deep water behaviors

Tsunami energy balls accelerating by (2.1): (a) keep long wave tsunami speeds, if startedat long wave speeds; (b) follow long wave tsunami ray paths; (c) conserve energy flux;and (d) obey the long wave shoaling relation (Green’s Law).

2.2 On land accelerations

On land, tsunami are not wave-like but flood-like–the distinction being that in the latter,energy and mass move at the same speed. On land, we assume initially that the tsunamiballs accelerate according to

al(t)= [g• s(p(t))]s(p(t))=g−[g•n(p(t))]n(p(t)). (2.7)

Unit vectors n(p) and s(p) are the upward unit normal and down slope directions of thetopography T(x,y) at p (Fig. 1, right). You calculate these directions from T(x,y) from

n(p)=−∇T(x,y)+ z

√

1+∇T(x,y)•∇T(x,y), (2.8)

s(p)=−z+(z•n(p))n(p)

√

1−(z•n(p))2. (2.9)

Unlike (2.1), (2.7) is meant to be a real equation of motion. Acceleration (2.7) is simply thecomponent of gravity resolved in the downslope direction of topography T(x,y). Becauseg• s(p(t))≥ 0, (2.7) always accelerates on-land tsunami balls downhill, back toward theocean.

2.3 Self-topography

Ward and Day (2006) used Eq. (2.7), with two friction terms, to model the Mt. St. Helen’sdebris avalanche. For water flows, additional self-topographic forces are desirable to


Figure 2: (Top) Self-topography on land. Downslope directions are not evaluated on the topographic surfaceT(x), but rather on T(x)+A(x,t), the on-land surface of the water. (Bottom) Self topography in shallowwater, downslope directions are evaluated on A(x,t). Wherever applied, self-topography pulls water piles apart.

mimic flooding. To include self-topography, replace land topography T(x,y) in (2.8) and(2.9) with

TS(x,y,t)=T(x,y)+A(x,y,t),

where A(x,y,t) is the water height at (x,y) at time t. A(x,y,t) relates to the numberdensity of tsunami balls σ(x,y,t) that happen to be at that position at that time through(2.5). Eq. (2.7) now resolves the acceleration of gravity in the downslope direction oftopography plus the water height (Fig. 2, top). TS(x,y,t) of course, describes the on-land water surface itself. Self-topography drives piles of water apart by acceleratingtsunami balls on opposing slopes outward (Fig. 2, top). Self-topography complicates thecalculation in that (2.8) and (2.9) become time-dependent.

2.4 Shallow water/breaking accelerations

If tsunami balls followed Eq. (2.1) all the way to shore, they would come to a completestop where H = 0 and never make it onto land. Clearly, between the deep water wave-like zone (2.1) and the on-land flood-like zone (2.7), tsunami transition. In shallow waterwhere wave heights surpass Acrit, we adopt a transitional shallow water acceleration

asw(t)=adw(t)+(1/2)[g• sw(p(t))]sw(p(t)), (2.10)


where

nw(p,t)=−∇W(x,y,t)+ z

√

1+∇W(x,y,t)•∇W(x,y,t), (2.11)

sw(p,t)=−z+(z•nw(p,t))nw(p,t)

√

1−(z•nw(p,t))2. (2.12)

Appropriately, (2.10) blends on land and deep water behaviors. The first term repli-cates the deep water acceleration (2.1). The second term is one-half of (2.7) includingonly the self-topographic forces imparted by slopes of the water surface (Fig. 2, bottom).Unlike (2.7), acceleration (2.10) acts only on tsunami balls in water so land topographyT(x,y) need not be included. Self-topographic forces imposed in shallow water are fun-damental to runup and inundation. Self-topographic forces accelerate the leading edgeof tsunami onto land despite the efforts of long wave forces (2.1) to stop it at the beach.Self-topographic forces also act to turn back trailing edges of tsunami prior to reachingthe beach. Wherever applied, self-topography flattens water piles. In water shallowerthan Acrit, waves subject to (2.10) are in fact, breaking.

2.5 Friction

Following Ward and Day (2006) we introduce a dynamic drag friction

aF(t)=−ν[v2(p(t))]v(p(t)), (2.13)

where ν = 0.001/m is a small, but adjustable coefficient. Frictional accelerations alwaysact in opposition to velocity. Dynamic drag (2.13) intends to represent an actual energyloss but it also helps to stabilize numerical realizations, particularly in restraining ballssent careening by errant evaluation of topographic slope. We add (2.13) to accelerations(2.7) for those tsunami balls on land and also to accelerations (2.1) for any deep waterballs that, for numerical reasons, have gone hyperspeed v(p)>

√

gH(p).

3 Calculations in two dimensions

3.1 Simple beach

To illustrate the tsunami ball approach to runup and inundation, consider a two-dimensional coastline with topography

T(x)=−H0(1−e−xtan(β)/H0) : x>0,

T(x)=−xtan(β) : x<0,(3.1)

where H0 is the deep water depth and β gauges the slope at the beach (Fig. 3). Onto thisshore, tsunami balls will be sent in from the right. In Fig. 3, the line labeled AG(x) traces


Figure 3: Geometry for two-dimensional runup and inundation. Wave pulses of different initial amplitudeand wavelength run to shore from the right. Topography (brown) follows Eq. (3.1). The two black linestrace Greens Law (3.2) and the proposed value of Acrit. Where wave height is less than Acrit, tsunami ballsaccelerate according to (2.1). When wave height exceeds Acrit, accelerations revert to (2.10). Tsunami ballson land (x<0) accelerate using (2.7).

Green’s Law

AG(x)= A0

[

H0

H(x)

]1/4

, (3.2)

with A0 being the wave amplitude in water depth H0, and H(x) a water depth closer toshore. We have established that waves of tsunami balls following deep water accelera-tions (2.1) grow according to this curve. AG(x) runs to infinity at the shore and beliesthe failure of (2.1) to take tsunami close to the beach. The line Acrit(x) in Fig. 3 marksthe critical wave height where tsunami balls transition from accelerations (2.1) to (2.10).Wave shoaling experiments (Le Mehaute and Wang, 1996) suggest that water waves de-viate from Green’s Law when their amplitude exceeds 3/4 to 5/4 of the local water depthH(x). In this article, we select a simple wave amplitude-based breaking criterion

Acrit(x)=H(x)

2(3.3)

to define the transition. (Coefficients different from 1/2 or other wave slope or bot-tom slope-based breaking criteria might be considered in the future.) Where AG(x)and Acrit(x) cross, waves may, if they are large enough, take on shallow water behav-ior (2.10). Recall that (2.10), unlike (2.1) includes accelerations due to self topography.Self-topography spreads the wave and cuts its height in a breaking action. The conse-quences of breaking depend upon the length of time that the force acts (proportional tothe distance offshore that lines AG(x) and Acrit(x) cross) and the strength of the force(proportional to the steepness of the wave). Not surprisingly, breaking actions are com-plex, being functionals of ocean depth along the wave path, wave period, wave ampli-tude and wave shape.

The following two-dimensional experiments distribute 16,000 tsunami balls with aGaussian number density σ(p) just off the right side of the figures. The width of the


Gaussian distribution varies in the different experiments to simulate waves of differentperiods. (In this paper, the period of a single wave crest is just the time it takes to passa point.) We select tsunami ball energy E0 to give the desired initial wave height as

computed by (2.5). The i-th ball starts with a leftward velocity of√

gh(xoi ) where xo

i is

its initial position. Once started with these initial conditions, Eqs. (2.1), (2.7), (2.10) and(2.13) guide the evolution of the wave for the remainder of the experiment. Note that therotation term R(t) vanishes in two-dimensions.

3.2 Shallow slope–short waves

To construct a representative coastal section, let H0 =100m and β=0.11◦ (1:520 slope) in(3.1). The first example (Fig. 4) sends in a single pulse of 50s duration and 20m height. At50s, the period of this pulse exceeds the 10-15s periods of wind driven waves. Tsunamiof 50s period might be sourced from small volume landslides or from impacts of aster-oids of about 100m diameter. With 20m amplitude, this steeply-sided wave might berepresentative of the latter.

At first (Frames 1 and 2, Fig. 4), tsunami balls obey deep water accelerations and waveamplitude follows Greens law (3.2). Near x=25km wave amplitude first tops Acrit(x) andthe self-topographic forces turn on and start to shave and spread the pulse. As picturedin Fig. 2b (bottom), self-topography acts to accelerate material on the leading edge ofthe wave toward shore and material on the trailing edge back to sea. As a consequence,after continued contact with Acrit(x), the single pulse begins to tear apart. For a periodsof time and portions of broken wave however, amplitudes fall below Acrit(x) and (2.1)again governs the tsunami balls temporarily (Frame 3). Wave breaking is not necessarilycontinuous. As the wave approaches shore, breaking eventually knocks down the singlepulse into several (Frames 5 and 6). Frame 7 shows maximum inundation of 8.7m -considerably less than the 20m input height. Breaking across shallow seas takes a heavytoll on steep sided, short wavelength tsunami. Runup amplification R/A0 (the ratio ofinitial height to runup height) in this instance is just 0.43.

The red line in Fig. 4 plots the peak value of wave height offshore, or peak flow depthplus topography onshore. Due to interfering in- and out-flows, peak runup might befound anywhere along the beach, not always at the point of maximum incursion. Wavebreaking and wave interference incorporate highly non-linear processes. Minor differ-ences in initial conditions spawn variations in runup even under idealized conditions.In real world applications where topographic variations are stronger and poorly known,”random” elements in runup increase (Section 4).

The reader should be aware that the runup and inundation figures in this article (plusothers) are accompanied by Quicktime movies. The movies reveal aspects of runup,breaking and inundation far better than the still frames. To view the Quicktime movie forFig. 4 click RU(50s-20m-0.11).mov (Note– prefix ”http://es.ucsc.edu/∼ward/” has beendropped from all of the hyperlinks in the text).


Figure 4: Runup and inundation of a 20m high, 50s duration wave. Large amplitude, short wavelength tsunamiwaves may break far offshore, reducing their runup potential.

3.3 Shallow slope–intermediate waves

Fig. 5 illustrates runup and inundation of a 20m high, 180s period wave pulse. Tsunamiof 180s period might be sourced from moderate volume landslides or from impacts ofasteroids of about 2000m diameter. A similar sequence of events affect the 180s pulseas did the 50s pulse in Fig. 4. The larger-dimensioned 180s wave pulse however, resistsdisruption more than shorter length 50s waves. Still, breaking takes its tax and peakrunup from the 20m wave reached only 12.6m. Runup amplification increased to 0.63from 0.43 for the 50s pulse, but R/H0 remains less than one. The Quicktime movie forFig. 5 can be seen by clicking RU(180s-20m-0.11).mov


Figure 5: Runup and inundation of a 20m high, 180s duration wave. A 180s wave period might be generatedfrom moderate volume landslides or from impacts of large asteroids.

3.4 Shallow slope–long waves

Fig. 6 illustrates runup and inundation of a 20m high, 1200s period wave pulse. Waveswith 10 to 20 minute period characterize tsunami generated by subduction zone earth-quakes or very large landslides. Breaking affects waves of this dimension differently.Contact with Acrit shaves wave height (Panels 2-5, Fig. 6) but otherwise only slightly dis-rupts its progress. A long period tsunami acts like a 100 car coal train–even after brakesare applied, a long time elapses before the train slows. Note the bore-like steep front thatdevelops as the wave approaches and onsets land (Panels 4-6). Waves of this length raise


Figure 6: Runup and inundation of a 20m high, 1200s duration wave. Typical of earthquake tsunami, longwaves like this are difficult to stop.

flood-like inundations lasting many tens of minutes. Peak runup now reaches 23.5m andR/H0 increased to 1.17 from 0.63 for the 180s pulse. Play the Quicktime movie for Fig. 6here RU(1200s-20m-0.11).mov

Figs. 4-6 argue that for tsunami pulses of fixed height running onto shallow beaches,breaking impacts longer period waves less than shorter period ones. A sensible proposalfor sure, but it would be useful to consult laboratory results for comparison and cali-bration. Unfortunately, no experiment that we are aware of employs slopes as shallow(1:500) nor waves as long (λ0 =10−500H0). In fact, rarely do wave tanks slope shallowerthan 1:50 or do input wavelengths exceed a few times H0. The need to extrapolate labo-


ratory experiments to ’outside’ scales has always loaded heavy uncertainly onto tsunamiresearch. (See Appendix A).

3.5 Steeper slope–long waves

Here, we increase beach slope to 0.5◦. A 1:114 incline is moderately steep for real worldsituations but it still falls at the shallowest fringe of lab experiments. For a fixed waveheight, steeper slopes bring the crossover between AG(x) and Acrit(x) closer to shore.Generally, the closer that the cross over point approaches shore, the shorter the time thatbreaking operates on the wave and the greater the final runup surge. In Fig. 7, the wavevirtually contracts the beach before breaking criterion (3.3) kicks in (Panel 4). Collapse ofthe steep front face (Panel 5) forces runup to 40.5m–double that of Fig. 6. The Quicktimemovie for Fig. 7 can be seen by clicking RU(1200s-20m-0.5).mov

3.6 Offshore pile-ups and reverse breakers

Curiously in these calculations, only a fraction of the tsunami balls sent toward land crossthe coastline. Many balls get ”ponded” behind the wave’s front piling on the beach. Thewater pile-up is analogous to traffic jamming at an obstruction, or sports fans queuingat turnstiles. Water in the rear of the wave actually turns back in reverse facing, thenreverse traveling breakers even before peak runup (Panels 4-6, Fig. 6 and Panels 6-7,Fig. 7). Few observations of near shore pile-up exist because most tsunami are witnessedfrom land where attention focuses upon onshore events. However, photos taken fromyachts anchored in 10 to 15m of water (about twice tsunami amplitude) offshore Phi PhiIsland Thailand during the December 26th 2004 tsunami document both the near shorewater pile-up and steep off shore-directed waves. A series of digital, time-stamped pho-tographs http://www.yachtaragorn.com/Thailand.htm provide flow indicators in theseaward-pointing bows of other anchored yachts and the bow waves generated by therapid motion of the water (We reproduce selected photos in Fig. 16, Appendix B). Duringthe initial period of onshore-directed flow in the 2004 tsunami, the photographs show thedevelopment of a steep, breaking front at the seaward side of a near shore pile-up of wa-ter several meters high. Over a period of a few minutes, while the overall flow directionwas still shoreward, this front developed into multiple reverse traveling breakers withlengths several times that of the yachts. One of the boats reported to have grounded on apatch reef during the initial withdrawal but, as if riding on the pile thereafter, it gave noreport of grounding in later wave troughs.

The yacht Mercator, anchored off Nai Harn beach, Phuket provides further evi-dence of the nearshore pile-up (See Fig. 17, Appendix B). The yacht’s echo sounderhttp://www.zeilen.com/publish/article 1659.shtml records: (a) pre-tsunami waterdepth of 11 to 12m; (b) the initial drawdown to a minimum water depth of 8m; (c)three following crests with maximum water depths of 15, 13 and 17 meters at intervalsof around 13 minutes. At no time between these crests did soundings drop below 11m.


Figure 7: Runup and inundation of a 20m high, 1200s duration wave on a steep 0.5◦ beach. On steep beachesbreaking has little time to operate so runup amplifications are large. Note the formation of reverse breakers inPanels 5-6.

Thus, for the first hour of the tsunami, the mean water depth offshore was significantlygreater (by ∼1.5m) than prior to the tsunami, again indicating a persistent pile-up inshallow water.

3.7 Conclusions about single wave runup in two-dimensions

Fig. 8 summarizes runup results for the examples shown and several others. The figureplots runup amplification (R/A0) versus input wave amplitude normalized by the initial


50m, 95m and 100m water depths of the experiments. (For other starting conditions

A′0 and H′

0 replace A0 on both axes by A′0 [H

′0/H0]

1/4.) Colors red, orange and green

display (R/A0) for 1200, 180 and 50s. In virtually all cases, runup amplification increaseswith wave period and decreases with initial wave amplitude. Simply put, breaking morestrongly affects steeper waves than flatter ones.

Figure 8: Summary of two-dimensional single wave runup experiments for fixed H0. (Left) 0.11◦ slope 1:520,H0 = 50m. (Middle) 0.5◦ slope 1:114, H0 = 95m. (Right) 1◦ slope 1:57, H0 = 100m. Red, Orange and Greencurves correspond to input waves of 1200s, 180s, and 50s period. Longer period waves on steeper beachesgenerally produce larger runup relative to their input amplitudes.

The dashed line in the Fig. 8 plots

R/H0 =(A0/H0)0.8 or R/A0 =(A0/H0)

−0.2, A0 < H0≪λ0. (3.4)

Ward (2002) and Chesley and Ward (2006) used (3.4) as a universal runup formula insituations where no case-specific information was available. They realized that Eq. (3.4),being independent of wave period or beach slope, can not hold everywhere (See Ap-pendix A). Eq. (3.4) always returns R/A0 >1, but we have already seen (e.g. Figs. 3 and4) that R/A0<1 when breaking is strong (large, short period waves on shallow beaches).On the other hand, Fig. 8 proposes that (3.4) understates runup in weakly breaking cases,so the formula might represent a decent compromise, given no other information.

Empirical runup formulas like (3.4) derive from solitary wave experiments that ignore”set up” (see below). Such formulas, even more complex ones (See Appendix A), broadbrush reality. Moreover, randomness in runup may swamp differences due to beachslope, wave period etc. In Section 5, we argue that run up is more practically viewedas a random variable. At best, empirical runup laws might characterize the mean of itsdistribution.


Figure 9: Runup and inundation of a set of twenty, 20m high, 180s duration waves. Note the slow and persistentflooding of the land due to set up. Set up may be more important to certain tsunami hazards than run up.

3.8 Shallow beach–sets of waves

Run up gets most of the attention in tsunami experiments, but tsunami hazard stems notjust from runup height, but run in distance too–that is, how far water penetrates inland.Run in depends on the steepness of the beach and the size and period of the tsunami.Run in also varies with the number of impinging waves.

Fig. 9 samples runup and inundation from a set of twenty 180s wave pulses like Fig. 5,sent into shore at five minute intervals. Near the coast, it takes time for the outflow ofwater to match the influx of water carried by the waves. We discussed the pile-up of


single waves in the section above. Now, with multiple waves, the pile-up amplifies insize and duration and the total effect is called ”set up”. Set up shows largely as a slowflooding of the land. Set up harks to storm surge, but set up is not driven by atmosphericpressure or wind. The duration of set up induced flooding spans not just the duration ofone wave, but the duration of the entire wave train. This might mean hours. The singlewave (Fig. 5) ran up 13m and ran in 7km whereas the wave set in Fig. 9 ran up 22m andran in 12km. Preliminary experiments suggest that in the presence of set up, long wavetrains produce relative runups R/A0 ∼ 1 [A0 < H0/2; H0 ∼ 100m] independent of beachslope or wave period. The additional run-in distance from a set of small waves mightbe more hazardous in terms of the number of folks affected than a single wave of higherrun up but smaller run in. Set up is especially relevant to impact- or landslide-generatedtsunami because these entail long trains of waves. Click here RU(180s-20m-0.11)set.movfor a Quicktime version of Fig. 9.

4 Calculations in three dimensions

The true beauty of tsunami balls show in three-dimensions where rotation of ball tra-jectories toward shallower water come into play. Rotation of wave fronts obliquely ap-proaching shore to a more normal incidence and refraction of energy around headlandsare common manifestations. Strong bending and repeated reflections from land also trapwave energy in offshore shallows.

4.1 Babi Island

Consider first, an island with circular symmetry (Fig. 10) modeled after Babi Island, In-donesia. In the Flores tsunami of 1992, waves refracted around the conical island volcanoand converged on its lee side. Exceptionally high runups there destroyed two villagesthat, at first thought, might have been protected by the island. Briggs et al. (1994, 1995),Liu et al. (1995), and Titov and Synolakis (1995, 1998) adopted Babi Island as an experi-mental and computational benchmark.

Our simulation adopts an ocean depth of the form

h(r)=310m−360me−r2 /30,000, (4.1)

where r measures distance from the Island’s center in meters. Eq. (4.1) generates an island20km wide with a slope at the beach of about 0.5◦. Toward this target we send 50,000tsunami balls in an initial uniform 4m high, 35km wide (600s period) wave as pictured in

the upper left panel of Fig. 10. The i-th ball starts with a rightward velocity of√

gH(ri0)

where ri0 is its initial position. What run up predictions do the two dimensional results

offer? Green’s Law (3.2) says that a wave starting at 310m depth should have amplitude4(310/95)0.25 = 5.4m at 95 m. Given an offshore height to offshore depth ratio there of


Figure 10: Application of tsunami ball theory to Babi Island runup and inundation. Note the focussing andcompression of the wave at the Island front and the wrap-around at the Island back. The yellow box at lowerleft shows the statistical distribution of runup heights along the coast.

∼0.06, a beach slope of ∼ 0.5◦, and a period of 600s, the center frame of Fig. 8 predictsthat on a plane beach, the wave should run up ∼3.1 times its height at 95m depth, or16.7m.

Panels 1-4 of Fig. 10 illustrate the sequence of wave compression, inundation, andreflection off Babi Island in a circular front. Panels 5-7 show wave wrap-around. Runupson the exposed front side of this island reached 16-18m, a bit more than expected fora plane beach, but refraction focuses energy toward the nose of the island. Runups onthe lee of the island reached 11-12m–less than the front side, but still significant. In themean, the 16.7m prediction holds well. Panel 8 shows multiply refracted/reflected wavesstarting the second orbit of the Island. A few tsunami balls get trapped for hours. Moviesof Fig. 10 are here Babi.mov and Babi-num.mov. The second of these samples runupheights and locations.


Figure 11: Northern California tsunami inundation simulation. This scenario supposes that a Cascadia mega-thrust earthquake sends a 4m wave landward from the trench. Note the strong reflections from the steepheadlands and the flooding of the lowlands. The yellow box at lower left shows the statistical distribution ofrunup heights along the coast. Bold contours are 500m.

4.2 Northern California–a Cascadia earthquake tsunami

In United States’ Pacific Northwest, a major tsunami hazard lurks in the Cascadia sub-duction zone just to the west of the continent. The next Cascadia mega-thrust earthquake(∼M9) might slip the zone 15-20m and uplift the seabed east of the trench by severalmeters. Fig. 11 pictures a 60,000 tsunami ball simulation of runup and inundation ofnorthern California for a plausible Cascadia earthquake scenario.

Like the Babi Island example, a uniform 4m high, 35km wide (360s period in this wa-ter depth) wave is sent toward the California coast. The i-th ball starts with a rightward

velocity of√

gH(ri0) where ri

0 is its initial position. To 100m depth, slope here measures

1:150 to 1:250. Green’s Law (3.2) says that a 4m wave starting at 1000m depth shouldhave amplitude 4(1000/95)0.25 =7.1m at 95m. Given an offshore height to offshore depthratio there of ∼0.07, a beach slope of ∼0.5◦, and a period of 360s, the center of Fig. 8 pre-dicts that on a plane beach the wave should run up ∼2.8 times its height at 95m depth,or 19.9m.


Unlike the Babi Island example, real offshore and onshore topography is ragged.Tsunami balls, accelerating in proportion to the gradient of topography, take on a ran-dom walk component. Even before striking shore, ray theory refraction of balls grazingsteep slopes generates streaks of concentrated and diffused energy (Panels 2-3, Fig. 11).Small changes in wave incident direction move these streaks around in an unpredictablefashion.

Upon beaching, those balls contacting steep topography bounce off in coherent re-flections whereas those balls meeting coastal flats or drainages rush inland. Many ofthe reflected balls become trapped in littoral drift, striking the coast repeatedly as theirpaths rotate toward shallow water. Some of the balls that rushed inland become stalledbehind low coastal hills and dunes. Bits of tsunami persist to six hours and more. Likemuch of California, the steep coastline here limits tsunami run-in to isolated flatlandsand drainages. Of course, major populations often center at such locations–e.g. Eurekaand Crescent City. If a Cascadia earthquake were to send in a 4m wave as modeled, wepredict runups between 4m and 27m with a mean of 15.4m. Full runup statistics (yellowbox inset of Fig. 11), in addition to likelihood estimates that a Cascadia earthquake wereto send in a 4m wave versus a 3m one, versus 2m one etc. form a sound basis for prob-abilistic tsunami hazard analysis. Click here Humbolt-in2x.mov and Humbolt-in2.movfor Quicktime versions of Fig. 11.

4.3 Southeast United States–A La Palma wave

Fig. 12 stages a runup and inundation simulation for the southeast United States. Thecoasts of Florida, Georgia and South Carolina present a contrasting tsunami environ-ment to Northern California. On one hand, the southeast coast has a wide and shallowcontinental shelf. Water sounds to 30m depth and less out to 60km from shore (1:2600slope). Judging from the two dimensional simulations, large steep waves in this environ-ment may break far from the beach. On the other hand, 10m elevations onshore are nottopped for 60km inland and many wide waterways deeply penetrate the coast. Tsunamithat do reach the shore face few obstacles to minimize run in.

No subduction zone exists off the east coast of the United States, so earthquaketsunami here are rare and small. Potential landslide tsunami sources however, do existin the Atlantic basin. Ward and Day (2000) present evidence that Cumbre Vieja Volcanoon La Palma Island is in the later stages of a growth/collapse cycle that will climax in aflank failure involving as much as 500 km3 of material. Their simulations predict that ahigh speed landslide of this volume might fling dozens of 20m high, 10 minute periodwaves toward the U.S. East Coast.

Fig. 12 pictures a 100,000 tsunami ball simulation of runup and inundation from asingle 10m high, 35km wide, 660s period, La Palma wave. The i-th ball starts with a

rightward velocity of√

gH(ri0) where ri

0 is its initial position. Green’s Law (3.2) says that

a 10m wave starting at 200m depth should have amplitude 10(200/50)0.25 =14m at 50m.


Figure 12: Runup and inundation of Southeast U.S.A. from a single 10m high, 660s period tsunami parentedby a La Palma Island collapse. Run in across the lowland extends 30-50km. Thin and bold contours offshoreare 10 m and 100m.

Given an offshore height to offshore depth ratio there of ∼0.28, a beach slope of ∼0.011◦,and a period of 660s, the left frame of Fig. 8 predicts run up ∼1.1 times wave height at50m depth, or 15.4m. Actual slopes (1:2600) fall considerably short of 1:520 so the 15.4mestimate is probably high. Wave breaking commences 60 km out and has 1-2 hours to act(Panel 2 Fig. 12). The wave spreads and some balls turn back before the wave reachesland (Panels 3-6). Note the lack of reflections, the flat coast acts like a tsunami absorber.When all is said and done, run up averaged 12.6m with a large spread. Run in across thelowland extends 30-50km.

Run up amplification in Fig. 12 hardly exceeded one, so the shallow shelf adjacent tothe Southeast U.S. does offer a modicum of protection from a 10m, 660s La Palma wave.Still, Ward and Day (2006) forecast a great sequence of waves from a La Palma landslide,not just one. Recalling the effect of wave set up (Fig. 9), likely flooding will be moreextensive than pictured. Click here Savanna-10.mov and here Savanna-10-num.mov fora Quicktime versions of Fig. 12.

5 Runup as a statistical quantity

The three dimensional simulations in Figs. 11 and 12 input waves of uniform height, yetrunup varied widely. The spread finds origin both offshore and onshore: (a) Peculiaritiesof offshore bathymetry concentrate or disperse tsunami energy in deep water. (b) Pe-culiarities in wave breaking toss tsunami balls onto land with different speeds and turn


Figure 13: (Top) Cumulative distribution of run up values from the simulations of Figs. 11 and 12 normalizedto their mean value. (Bottom) 2004 observed normalized runup distributions from four sites in Thailand andSumatra. Normalized runup may have the same statistical behavior everywhere.

back others. (c) Peculiarities of onshore topography funnel water or overlap reflections.(d) Of course too, some shoreline sites are just more sheltered than others. Even in thesefully deterministic calculations, minor changes in initial wave conditions play throughto substantial changes in runup locally. Because such variations also occur in nature,constructing single valued inundation hazard maps is fruitless. We believe that the onlymeaningful description of runup is in terms of a random process. The best that one canhope for in describing a random process is to characterize its statistical properties. Ofthese, mean and variance about the mean head the list.

Fig. 13 (top) plots cumulative distribution of runup heights from the Northern Cali-fornia and Southeast U.S. inundation simulations normalized to their mean values. Al-though conditions for the two cases were markedly different, both distributions trendsimilarly–spreading to about 1/2 to 2 times the mean value. Fig. 13 (bottom) plots cumu-lative probability of runup height from the 2004 Sumatra tsunami at Banda Aceh, WestSumatra, Phuket and Khao Lak as summarized from post-event surveys (Matsutomi etal., 2005; Sato et al., 2005; Tsuji et al., 2005). These four locations too, separate widelygeographically and in mean wave height, but the normalized run up distributions nearlyoverlap. Although our analysis is cursory, we speculate that normalized runup may havethe same statistical behavior everywhere. If so, a single estimate of mean would fullycharacterize runup statistics. Such a finding would greatly simplify probabilistic hazardestimation by obviating the need for detailed tsunami simulations at every location.


6 Conclusions

This paper develops a new, granular approach to the ”last mile” problem of tsunamirunup and inundation. The grains employed here are not fluid, but bits, or balls, oftsunami energy. By careful formulation of the ball accelerations, both wave-like andflood-like behaviors are accommodated so tsunami waves can be run seamlessly fromdeep water, through wave breaking, to the final surge onto shore and back again. In deepwater, tsunami balls track according long wave ray theory. On land, tsunami balls be-have like a water landslide. In shallow water, the balls embody both deep water and onland elements. Tsunami balls offer a competitive alternative to traditional run up models.The tsunami ball approach applies to any environment (defended harbors, lakes, fiords)where non-dispersive ray theory describes deep water wave propagation.

The introduction of self-topography to tsunami ball accelerations mimics flood-likebehaviors on land. Off shore, self-topography induces breaking that limits subsequentinundation by disrupting inbound waves. Consequences of breaking are complex anddepend on the wave period, wave amplitude, wave shape and beach slope.

When applied to two-dimensional single waves of 50-1100s period on 1:100-1:500beaches, we find that runup amplification increases with beach slope and wave periodand decreases with input wave amplitude. These trends coincide with laboratory results,but few wave tank experiments approach the low slopes and long wavelengths neededto make a quantitative comparison.

Tsunami ball simulations predict that only a portion of the wave sent to the beachactually reaches shore. Much of the water piles up, then returns to sea as reverse break-ers prior to peak run up. When tsunami consist of many waves, the pile-up grows induration and size and slowly floods the land as set up.

When applied to three-dimensional geometries, we find that the two-dimensional re-sults generally hold, although runup values widely scatter due to complications of waverefraction and interference. Two cases employing real topography–a steep Californiacoast and a flat Georgia coast–highlight the practical elegance of the tsunami ball ap-proach. A 4m wave sent onto the California coast from a Cascadia earthquake averages15.4m runup. In contrast, a larger 10m La Palma wave directed to the Georgia coast runsup just 12.7m on average. Georgia’s adjacent wide shallow continental shelf provides abit of protection from steep, large amplitude tsunami in the form of a lower runup am-plification. Conversely, Georgia’s flat coastal terrain and penetrating watercourses offerno resistance to run in, once tsunami reach shore.

Because of their highly non-linear nature, we argue that runup and inundation arebest considered to be random processes rather than deterministic ones. Models and ob-servations hint that for uniform input wave conditions, normalized runup statistics ev-erywhere follow a single skewed distribution with a spread between 1/2 and 2 times itsmean.


Figure 14: Summary of tsunami ball runup experiments in Fig. 8 weighted by tanβ to the power q. Exponentq decreases with increasing wave period.

Appendix A: Separation of slope and period effects on the runup

of single waves

By plotting the results of Fig. 8 weighted by (tanβ)q, the influences of beach slope andwave period on the runup of single waves separate nicely (Fig. 14). Best-fit q-exponentsdecrease consistently with increasing wave period. Moreover, the uniform increase inslope of the least square fits in Fig. 14 suggest a runup law of the form

R

H0=C(T)

[

(tanβ)q(T) A0

H0

]p(T)

(A.1)

or

R

A0=C(T)(tanβ)q(T)p(T)

[

A0

H0

]p(T)−1

, (A.2)

with

C(50)=3.26; q(50)=1.02; p(50)=0.41;

C(180)=3.93; q(180)=0.65; p(180)=0.53;

C(1200)=3.52; q(1200)=0.24; p(1200)=0.73.

Fig. 15 plots (A.1) for the three periods above for slopes of 1◦ (top) and 2.88◦ (bottom).The latter 1:19.85 slope, three times steeper than any case examined here, corresponds tothe shallowest slope experiment run by Synolakis (1986). The dots in the bottom panelindicate relative runup heights of breaking solitary waves measured by Synolakis (1986)


Figure 15: Relative runup predicted for beach slopes of 1◦ and 2.88◦ for three wave periods. Experimentalobservations of breaking solitary wave runup are shown by the dots in the lower panel.

as reported in Li and Raichlen (2002). Solitary waves in the Synolakis experiments havecross sectional shapes

A(x)=(A0/H0)sech2

[

√

3A0

4H0x

]

(A.3)

and pulse widths of about 5/(A0/H0)1/2 times the offshore depth. In our experimentsH0=100m, so the equivalent wave widths of (A.2) have durations (15.8)(100m)/(9.8m/s2

100m)1/2∼50s for (A0/H0)=0.1 and (7.1)(100m)/(9.8m/s2 100m)1/2∼23s for (A0/H0)=0.5.Roughly then, the red line in the bottom panel of Fig. 15, corresponds to the wave lengthsof the Synolakis (1986) experiment. As yet, we have made no specific attempt to ”tune”the tsunami ball process to reproduce laboratory data. Judging from the fairly good fitin Fig. 15, the tsunami ball approach even as it is now, seems to capture many of thebreaking wave behaviors exposed in experiment.

Lastly, note that runups for the 1200s waves are only weakly dependent on beachslope in the range (0.11◦ to 1◦). A simplified runup formula for long period earthquaketsunami waves that ignores beach slope variations might take the form

R/H0 =1.5(A0/H0)0.7 or R/A0 =1.5(A0/H0)

−0.3,A0 < (H0/2)≪λ0; H0 =100m.

(A.4)

Contrast (A.4) with (3.4). Eqs. (A.1), (A.2) and (A.4) apply to 100m water depths and A0<

H0/2. For other starting conditions A′0, H′

0 and A′0 < H′

0/2, replace A0 by A′0 [H

′0/H0]

1/4.


Figure 16: Frames taken from a yacht at anchor off Phi Phi Island Thailand during the initial on shore flowstage of the December 26th 2004 tsunami. Note the reverse facing and traveling breakers turned back from anear shore water pile up similar to that predicted here http://es.ucsc.edu/∼ward/RU(1200s-20m-0.5).mov See-http://www.yachtaragorn.com/Thailand.htm

Figure 17: Depth Gauge recording from the ship Mercator. After the initial draw down, several wave crestsand troughs came by, but the water depth never fell below the pre-tsunami level. We believe that the ship wasriding on a water pileup. From: http://www.zeilen.com/publish/article 1659.shtml


Below are several 2-D and 3-D runup and inundation movies computed usingtsunami ball theory.

In this article:

http://es.ucsc.edu/∼ward/RU(50s-20m-0.11).mov




http://es.ucsc.edu/∼ward/RU(180s-20m-0.11)set.mov

http://es.ucsc.edu/∼ward/Babi.mov

http://es.ucsc.edu/∼ward/Babi-num.mov

http://es.ucsc.edu/∼ward/Humbolt-in2x.mov

http://es.ucsc.edu/∼ward/Humbolt-in2.mov

http://es.ucsc.edu/∼ward/Savanna-10.mov

http://es.ucsc.edu/∼ward/Savanna-10-num.mov

Others:


http://es.ucsc.edu/∼ward/RU(25s-5m-0.11+3).mov

http://es.ucsc.edu/∼ward/RU(25s-5m-0.11-3).mov







http://es.ucsc.edu/∼ward/Lituya-2D.mov

http://es.ucsc.edu/∼ward/shoreline-2.mov

http://es.ucsc.edu/∼ward/mont-bay.mov

http://es.ucsc.edu/∼ward/SFO.mov

http://es.ucsc.edu/∼ward/c-cod-final.mov

References

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[5] M. Herrmann, Two-phase flow, 2006 CTR summer program tutorial, 2006,http://www.stanford.edu/group/ctr/

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[13] D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 35(1966), 321-330.

[14] S. Sato et al., 2004 Sumatra Tsunami, Survey around South Part of SriLanka, 2005,http://www.drs.dpri.kyoto-u.ac.jp/sumatra/srilanka-ut/SriLanka UTeng.html

[15] C. E. Synolakis, The run up of solitary waves, J. Fluid Mech., 185 (1986), 523-545.[16] C. E. Synolakis and E. N. Bernard, Tsunami science before and beyond Boxing Day 2004,

Philos. T. Roy. Soc., 364 (2006), 2231-2265.[17] V. V. Titov and C. E. Synolakis, Modeling of breaking and non-breaking long-wave evolution

and runup using VTCS-2, J. Waterw. Port C.-ACSE, 121 (1995), 308-316.[18] V. V. Titov and C. E. Synolakis, Numerical modeling of tidal wave runup, J. Waterw. Port

C.-ACSE, 124 (1998), 157-171.[19] Y. Tsuji et al., The 26 December 2004 Indian Ocean Tsunami: Initial Findings from Sumatra,

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tation of debris avalanche deposits and landslide seismic signals of Mount St. Helens, May18th 1980, Geophys. J. Int., 167 (2006), 991-1004.

tsunamiballs: agranularapproachtotsunamirunup …ward/papers/tsunami_balls.pdf2 tsunami balls we...

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